Why does definite integral have the operation subtraction? Does the definite integral always have to involve finding the area and if not, why?

Question

For example

f' or f prime = [f(b) - f(a)] ÷ (b - a) = The integral from a to b

and f'(x)dx or f prime dx Divided by (b - a)

No area is being found. I hope there are better examples. I'm sorry

Answer 1

Although an easy way of thinking of an integral is with regards to the area under a curve, the integral is a notation that means nothing more than an infinite sum. The way we evaluate a definite integral happens to be done using subtraction in accordance with the 2nd Fundamental Theorem of Calculus:

If f(x) is a continuous function and F(x) is the antiderivative of f(x), meaning that F'(x) = f(x), then:

Answer 2

definite integrals are also used to calculate the length of a curved line, a one dimensional measurement, which is not an area, not a 2 dimensional measurement.

Length of a curve = from a to b = ∫√[1 + (dy/dx)^2]dx evaluated from a to b

your f' example involves a derivative, not an integral